Towards the Koch Snowflake Fractal Billiard

Contemporary Mathematics Volume 517, 2010

Towards the Koch Snowflake Fractal Billiard: Computer Experiments and Mathematical Conjectures Michel L. Lapidus and Robert G. Niemeyer Abstract. In this paper, we attempt to define and understand the orbits of the Koch snowflake fractal billiard KS. This is a priori a very difficult problem because ∂(KS), the snowflake curve boundary of KS, is nowhere differentiable, making it impossible to apply the usual law of reflection at any point of the boundary of the billiard table. Consequently, we view the prefractal billiards KSn (naturally approximating KS from the inside) as rational polygonal billiards and examine the corresponding flat surfaces of KSn , denoted by SKSn . In order to develop a clearer picture of what may possibly be happening on the billiard KS, we simulate billiard trajectories on KSn (at first, for a fixed n ≥ 0). Such computer experiments provide us with a wealth of questions and lead us to formulate conjectures about the existence and the geometric properties of periodic orbits of KS and detail a possible plan on how to prove such conjectures.

1. Introduction The long-term goal of this work is to justify the existence and investigate, via computer-aided experiments and suitable mathematical arguments, a prototypical fractal billiard, namely, the Koch snowflake billiard. Since the Koch snowflake curve (the boundary of the Koch snowflake billiard table, see Fig. 1) is nowhere differentiable, it is certainly not clear from the outset that the Koch snowflake billiard—or, let alone, the associated billiard flow—is a legitimate mathematical object of study. Indeed, the first question that springs to mind is “How do you reflect a billiard ball off of a nondifferentiable boundary?” On the other hand, because the Koch curve is highly symmetric (indeed, it is selfsimilar ) and can be thought of as some kind of ‘infinite polygon’ (see Fig. 1), it may not be entirely unreasonable to harbor such an expectation. 1991 Mathematics Subject Classification. Primary 37D40, 37D50, 37C27, 65D18, 65P99; Secondary 37A99, 37C55, 58A99, 74H99. Key words and phrases. Fractal billiards, Koch snowflake billiard, rational polygonal billiards, prefractal polygonal billiards, billiard flow, geodesic flow, flat surface, periodic (and quasiperiodic) orbits, Fagnano (and piecewise Fagnano) orbits, dynamical systems, fractal geometry, self-similarity, experimental mathematics, computer-aided experiments, mathematical conjectures and open problems. The research of the first author was supported in part by the U.S. National Science Foundation Grant DMS-0707524. c 2010 M. L. Lapidus and R. G. Niemeyer c 0000 (copyright holder)

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MICHEL L. LAPIDUS AND ROBERT G. NIEMEYER

In this paper, we provide experimental and geometric evidence towards this claim, as well as formulate conjectures and open problems concerning this challenging problem. In a nutshell, the Koch snowflake billiard is viewed as a suitable limit of (rational) polygonal billiard approximations; and analogously for the associated billiard (and geodesic) flows. We also identify a variety of (potential) periodic and quasiperiodic orbits of the Koch snowflake fractal billiard, and discuss some of their properties. In order to present our experimental results and conjectures about the snowflake billiard, we need to briefly recall several concepts from the theory of dynamical systems (more specifically, of rational polygonal billiards) and fractal geometry. A polygonal billiard B is given by a polygon P and the bounded region enclosed by P . For example, the square billiard is given by the square and the region bounded by the square. When studying a polygonal billiard, one is primarily concerned with the orbits of a pointmass moving at unit speed in B. We assume that our billiard ball experiences no friction and perfectly elastic collisions with the smooth portions of the boundary. Upon colliding with a smooth portion of the boundary, we reflect at the boundary according to Snell’s Law, which says that the angle of incidence equals the angle of reflection; see, e.g., [GaStVo], [Gu], Chapter 9 of [KaHa], or [Ta]. In general, if the billiard ball collides with a corner of the polygonal billiard table B, we must terminate the trajectory because we can not determine reflection at such points.1 We call the map that describes the flow on the billiard P the billiard map. An important case of a polygonal billiard is a rational billiard and is defined to be a polygon P for which every interior angle is a rational multiple of π; see, e.g., [MaTa], [Sm], [Vo] or [Zo]. Although a fractal is a shape that lacks many of the appealing qualities of polygons, so long as we are dealing with the right fractal, it is a shape that can be generated by polygons, a fact that is heavily exploited in this paper. The type of fractals we are interested in are self-similar fractal curves in R2 that are nowhere differentiable and have infinite length. Specifically, the fractal billiard that we eventually want to define and investigate is the Koch snowflake billiard, denoted by KS; the boundary of the associated billiard table is the Koch snowflake curve, denoted by ∂(KS). The construction of the Koch snowflake fractal curve is given in Fig. 1. The prefractal KSn of KS is then an nth level polygonal approximation of KS. Like many fractals, the Koch snowflake curve ∂(KS) has a non-integer dimension. In the case of the snowflake, that dimension is log3 4, which indicates that the curve ∂(KS) has infinite length. The Koch snowflake is not a rational polygon. The boundary of the Koch snowflake ‘fractal billiard’ is nondifferentiable, hence making it very difficult to properly define, and let alone analyze, the billiard map. However, the behavior of a pointmass may be anything but random, because, as was alluded to just above, KS has the very special property that there is a sequence of finite polygonal approximations {KSn }∞ n=0 converging to KS such that for each finite n, KSn is a rational billiard; see Fig. 1. In order to overcome the limitations of the Koch snowflake, we may examine the prefractal billiard approximations KSn and attempt to make an argument in support of the existence of periodic orbits of the limiting fractal billiard KS, based on results concerning the rational polygonal billiards KSn . Therefore, in 1We will see that there are polygons for which we can determine reflection at a corner, the equilateral triangle billiard being one.


TOWARDS THE KOCH SNOWFLAKE FRACTAL BILLIARD

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Figure 1. Construction of the Koch snowflake from the equilateral triangle Δ (with sides of length 1); from left to right: KS0 = Δ, KS1 , KS2 , KS3 . The process continues ad infinitum, yielding the Koch snowflake curve, viewed here as the boundary of the Koch snowflake billiard table KS. For each integer n ≥ 0, KSn denotes the nth prefractal polygonal approximation to KS. Note that KSn defines a rational billiard because its interior angles are either π/3 or 4π/3. While this property is not required to be a rational billiard, the interior angles of KSn alternate between π/3 and 4π/3. (Depending on the context, KSn also sometimes refers to the polygonal boundary of this rational billiard; the latter boundary is more correctly denoted by ∂(KSn ), however.) some sense, the snowflake curve is viewed as an (infinite) ‘fractal rational polygon’ and the associated billiard table KS as a ‘fractal rational billiard’. Roughly speaking, a compact set F ⊆ R2 is said to be self-similar if it can be written as a finite union of scaled (or rather, similar) copies of itself. More specifically, this means that F is a nonempty compact subset of R2 and there exists a finite collection {Wi }li=1 of contractive similarity transformations of R2 , with l ≥ 2, such that F = li=1 Wi (F ). It then follows that for any nonempty compact subset X of R2 , we have that limk→∞ W k (X) = F (in the sense of the Hausdorff metric), where W(X) := li=1 Wi (X) and W k denotes the kth iterate of W; see, e.g., Chapter 9 of [Fa]. For this reason, the self-similar set F is called the attractor of the iterated function system {Wi }li=1 . The Koch curve is a perfect example of a self-similar set; see Fig. 2.2 Strictly speaking, the Koch snowflake fractal is not self-similar; specifically, it is the result of pasting together three copies of the Koch curve; see Fig. 3. In light of this, we make a slight abuse of language by referring to KS (or rather, its boundary ∂(KS)) as a self-similar fractal. We can exploit the fact that the proposed billiard KS has a self-similar boundary. We notice that, in a sense, KSn is constructed from equilateral triangles. The billiard ball dynamics on the equilateral triangle billiard Δ = KS0 are well understood. In particular, the periodic orbits of the equilateral triangle billiard Δ are classified in [BaUm]. The resulting theorem, combined with the fact that KSn can be embedded in a tiling TΔn of the plane by equilateral triangles with side lengths 1/3n , will aid us in explaining our experimental results and formulating some of our conjectures. 2See also [Fa, pp. xiv–xv] and, in a related context, [La1,2], [LaNRG], [LaPa] or [La-vF, §12.3.1 & §12.5].


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Figure 2. Construction of the von Koch curve (left), often simply called the “Koch curve” here. At each stage of the construction, the middle third of each line segment is replaced by the other two sides of an equilateral triangle based on that segment. The self-similarity of the von Koch curve (right). The Koch curve is decomposed as the union of four pieces similar to the whole curve.

Figure 3. The Koch snowflake curve is obtained by pasting together three copies of the von Koch curve (from Fig. 2).

This paper is structured as follows. In Section 2, we discuss the necessary theory of rational billiards so that we may better present our results on the prefractal polygonal billiards KSn ; see Fig. 1. In §2.1, we discuss a useful tool for discerning periodic orbits of rational billiards, namely, the unfolding of billiard trajectories. In general, such an unfolding gives rise to a surface that depends on the orbit, but we also state known results for constructing a surface SP which is independent of the periodic orbit of the rational billiard P . We recall in §2.2 that the flow on the rational billiard P corresponds to the geodesic flow on the corresponding surface SP . In fact, as is explained in §2.3, vertices constitute singularities of the billiard map and certain copies of the vertices in SP constitute singularities of the flow on the surface. We discuss what implications this has for the billiard ball dynamics. In §2.4, we view the prefractal polygonal billiard KSn as a rational billiard, and discuss the consequences of this fact for the billiard flow at the vertices of KSn . We also construct the corresponding surface for KS1 , KS2 and KS3 and indicate the singularities of each surface. We show that, in agreement with the general theory of rational billiards, reflection at acute angles can be defined, but that reflection at obtuse angles cannot be determined unambiguously. In the last part of Section 2, namely §2.5, we discuss the results of [BaUm] on the equilateral triangle billiard and their implications for the billiard KSn . In doing so, we explain how KSn can be embedded in a tiling TΔn of the plane by equilateral triangles with side lengths 1/3n (as was alluded to above) and give an equivalence relation on the collection of all periodic orbits of the rational billiard Δ. We do not seek to generalize the results of [BaUm] to KSn but instead use these results to provide us with a list of appropriately adjusted initial conditions for testing in KSn .



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In Section 3, we present and discuss our experimental results on the prefractal billiard KSn by raising and sometimes answering a variety of research questions. In general, we examine the behavior of what we call induced orbits of KSn . An induced orbit is one for which the initial condition was appropriately adapted from an initial condition in the equilateral triangle billiard Δ. In §3.1, we examine the behavior of periodic orbits of KSn in the collection γn of billiard paths determined by the initial conditions (xmid , π/3), where xmid is a midpoint of a side of KSn and π/3 is the initial angle of the periodic orbit at the starting point xmid . We attempt to organize the orbits in γn according to their geometrical or dynamical behavior, and provide definitions for what we call Fagnano orbits, piecewise Fagnano orbits and primary piecewise Fagnano orbits of KSn . While these are very interesting orbits, the elements in γn do not constitute all of the induced orbits of KSn . In §3.2, we examine the behavior of periodic orbits induced by the initial condition (x0 , θ0 ), where x0 = xmid or θ0 = π/3 and the orbit remains nonsingular, meaning that it does not hit any vertices of the billiard table. A particularly interesting example of such a periodic orbit is given by (xmid + δx, π/6), where δx is a suitable value (or vector) used to perturb the initial basepoint lying at the midpoint xmid of a side of KSn . Because of the nature of the equilateral triangle billiard, the reflection at the vertices of Δ = KS0 can be determined. Consequently, in §3.3, these singular (and periodic) orbits are then used to induce singular orbits of KSn , for some n ≥ 0. Finally, in §3.4, we close Section 3 by discussing a seemingly uninteresting type of orbit, which we call a quasiperiodic orbit of KSn . Such an orbit can be considered as a ‘rational approximation’ to a given periodic orbit. For example, if (x0 , θ0 ) is an initial condition of a periodic orbit, then a corresponding quasiperiodic orbit would have an initial condition (x1 , a/b), where a/b is a rational approximation of θ0 (obtained via a continued fraction expansion of θ0 ). Because the boundary of the billiard KSn is changing as n increases, the study of quasiperiodic orbits of KSn should enable us to define a suitable notion of ‘quasiperiodic orbit’ of KS. In Section 4, we conclude our paper by stating several conjectures based on our experimental results and provide a list of open problems and conjectures related to the proposed fractal billiard KS and a particular collection of its periodic orbits. In particular, we conjecture the existence of what we call the primary piecewise Fagnano orbit of KS, which we propose would be a suitable limit of primary piecewise Fagnano orbits of the prefractal approximations KSn . In the special case of ppF , the ‘primary piecewise Fagnano’ orbit of KS,3 we also conjecture that its ‘footprint’4 on the boundary ∂(KS) is a self-similar subset of ∂(KS) ⊆ R2 ; more specifically, it is the natural counterpart of the middle-third Cantor set strung around the Koch snowflake curve ∂(KS). The other conjectures and open problems stated in Section 4 are clearly of a longer term nature. They concern, for example, the existence of the Koch snowflake

3The presumed periodic orbit ppF of KS is induced by the Fagnano orbit of the original

triangle Δ = KS0 (i.e., the shortest periodic orbit of Δ), along with its appropriate counterpart in each billiard table KSn ; see §3.1, along with Conjectures 4.4 and 4.5 of Section 4. 4 I.e., the subset of the boundary ∂(KS) consisting of all incidence points of the periodic orbit ppF of KS.


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billiard KS (as a proper mathematical object) and of the associated billiard flow,5 or (in the very long-term) the relationship between the length spectrum of the elusive Koch snowflake fractal billiard and the frequency spectrum of the corresponding Koch snowflake drum (e.g., [La1–2,LaNRG,LaPa], along with §12.3 and §12.5 of [La-vF]).6 In closing this introduction, we mention that the subject of billiards—and particularly, polygonal billiards or even, rational billiards—is an active area of research in the field of dynamical systems. Books, survey articles and research papers on various aspects of this topic include, respectively, [KaHa,Ta], [GaStVo,Gu,HuSc], [MaTa,Sm,Vo,Zo] and [BaUm,GuJu1–2,KaZe,Ma,Ve1–3]. To our knowledge, none of these references deals with the elusive case of ‘fractal billiards’. It is certainly true, however, that these works (and the many relevant references therein) have played a key role in motivating and guiding our investigations on the ‘Koch snowflake billiard’ (and its rational polygonal approximations) reported in the present paper. 2. Flows on Rational Billiards and Corresponding Surfaces In this section, we want to discuss the properties of the billiard flow on KSn and describe how KSn can be viewed naturally as a rational billiard. A rational polygon is a polygon with interior angles that are rational multiples of π. If the boundary of a polygonal billiard is a rational polygon, then we call the associated billiard a rational billiard. For us to discuss the flow on the associated closed surface, we need to define a few concepts.7 Definition 2.1. If H is a regular N -gon, where N is a positive integer, then DN is the group of symmetries of H and has cardinality 2N . It is called the dihedral group and is generated by the reflections in the lines through the origin that meet at angles π/N . Definition 2.2. Let A(P ) be the group of planar motions generated by the reflections in the sides of a polygon P . Furthermore, let G(P ) denote the subgroup of the orthogonal group O(2) consisting of the linear parts of the elements of A(P ). When the polygon P has a connected boundary, then we can give an alternate characterization of it being rational. Proposition 2.3. Let P be a polygon with connected boundary. Then P is a rational billiard if and only if the group G(P ) is finite. In that case, if the interior angles of the rational polygon P are written in the form π(mj /nj ) for j = 1, ..., r, where mj , nj are coprime positive integers and r is the number of vertices of P , then G(P ) is isomorphic to the dihedral group DN , where N is the least common multiple of n1 , n2 , ..., nr . If P is a rational billiard, then we have at our disposal a method for discerning periodic orbits. This method was first given in [KaZe] and reiterated in various 5as well as of the associated ‘fractal surface’ S KS and of the corresponding geodesic flow (conjecturally equivalent to the billiard flow on KS); see §2.2 and §2.4, along with parts (i) and (iii) of Conjecture 4.7. 6 See Open Problem 4.8. 7For an introduction to the theory of rational polygonal billiards from various points of view, we refer, e.g., to [GaStVo], [Gu], [MaTa], [Sm], [Ta], [Vo] or [Zo].



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Figure 4. Unfolding the equilateral triangle billiard: the orbit unfolds to a straight line segment (denoted by the dashed line). The terminal point on the upper right equilateral triangle in the unfolding corresponds to the initial point and the angles are the same. Hence, this corresponds to a periodic orbit of the equilateral triangle.

forms in the literature; see, e.g., the survey articles [GaStVo], [Gu], [MaTa] and [Zo]. Let x0 be the initial position of the billiard ball and θ0 be the initial direction of the billiard ball. Let x1 be the point on the side s1 of the billiard table at which the billiard ball collides, and θ1 be the angle at which the billiard ball reflects off of s1 . If the orbit α is periodic, then we want to illustrate this by ‘unfolding’ the billiard table and its contents. 2.1. Unfolding the billiard table B. To unfold the billiard table and its contents, reflect the contents of B in the side s1 . This results in a segment continuing in the direction of θ0 . Moreover, we see that we have created a situation where the trajectory passes through s1 and continues on to a segment that is collinear with the initial segment of the trajectory (see Fig. 4 for an example in the equilateral triangle billiard). Continuing this process, we see that we can unfold a billiard table and its contents. When the orbit is a collection of line segments, such an unfolding will be a straight line. Furthermore, the length of the unfolding is exactly the length of the orbit in B. When the orbit unfolds and terminates on an edge of a polygon P˜ in the unfolding such that the terminal point is a copy of the initial point and the segment makes an angle with the side of P˜ equal to the initial angle, then the original orbit of P is periodic. It is significant that we can unfold an orbit to determine the periodicity of the said orbit. In order to construct a closed surface out of the unfolding (which is determined by the periodic orbit), we identify sides of the unfolding based on where the ball collides. In particular, we identify the terminal side with the initial side since the orientation of P˜ and the original polygon P are the same. This results in


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a closed surface8 whose geodesic flow9 corresponds to the billiard flow on P . We can see this clearly in Fig. 4. However, this closed surface depends on the initial position and angle. We want to construct a surface S = SP that is independent of these initial conditions. In addition, the geodesic flow on that surface SP will turn out to be equivalent to the original billiard flow on P . 2.2. The invariant surface SP and its associated flow. Consider the product P × A(P ). Essentially, P × A(P ) contains 2N copies of our rational billiard table P . A graphical representation of P × A(P ) can be given by the following. If we fixed a vertex of a polygon P such that reflection in the adjacent sides of P generates 2N copies of P , then the resulting shape is what is called a generalized polygon. If two sides of the generalized polygon are translates of each other and/or the result of a rotation by π, then we consider these two sides to be equivalent. Then, upon modding out by this equivalence relation, we obtain a closed surface SP := P × A(P )/ ∼ that does not depend on any particular orbit. Example 2.4. Consider the triangular billiard P with boundary given by the triangle with interior angles (3π/8, π/8, π/2). We reflect P in the sides emanating from the vertex with angle π/8. Then, as is expected, we get a surface with 16 copies of the billiard P ; indeed, N = lcm{8, 8, 2} = 8 and so 2N = 16. Therefore, the associated generalized polygon is an octagon. Two sides are equivalent if they are translates of each other. Modding out by this equivalence relation, we have obtained the associated flat surface SP as shown in Fig. 5. Note, however, that since its genus is greater than 1, this surface cannot be properly embedded in the plane, but that for convenience, we represent SP as though it could be done; see Fig. 5. The surface SP is often referred to as a ‘translation surface’ or a ‘flat surface’ in the literature; see, e.g., [Ve3], [HuSc], [MaTa], [Vo] and [Zo]. Indeed, the geodesic flow on SP is nothing but the straight-line flow in the associated generalized polygon, with pairs of opposite sides identified as indicated above. The key fact concerning the geodesic flow on SP is that it is (dynamically) equivalent to the billiard flow on P . 2.3. Singularities of the billiard flow on P and of the geodesic flow on SP . As mentioned before, the billiard map is only well defined at certain types of vertices of the rational polygonal billiard. We demonstrate under which conditions the billiard flow can be defined at vertices. If P is a polygon, then the corresponding surface contains copies of the vertices of P . These copies of the vertices are then considered singularities of the flow on the surface and are called conic singularities. In order to understand what a conic singularity is, we must first introduce the notion of a conic angle. A conic angle is an angle that measures the radians required to form a closed loop about the origin. In the plane, the conic angle is 2π. We can form spaces in which the conic angle is not 2π. Suppose we had a space in which the negative upper half plane was glued to the negative lower half plane and the positive lower half plane was glued to the positive upper half plane of a second copy of R2 . Suppose we make similar identifications but instead glue the positive lower half plane to the positive upper half plane of the original copy of R2 . Then we 8By ‘closed’, we mean that the surface does not have any boundary. 9which is actually a straight-line flow on the closed surface



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Figure 5. The closed surface SP corresponding to the triangle P (3π/8, π/8, π/2); the associated generalized polygon is an octagon. Pairs of appropriate sides should be identified as indicated, in order to obtain SP , a surface of genus g = 2. Note, however, that strictly speaking, this planar representation is not valid. We illustrate such an embedding here as a heuristic device. have formed a space in which the conic angle about the origin is 4π. Consequently, there are two types of conic singularities: removable and nonremovable. A conic singularity is called removable if its conic angle is 2π. The conic singularity is called nonremovable if its conic angle is 2mπ, for some integer m ≥ 2. In the context of rational polygonal billiards, we determine the conic singularity of a vertex as follows. Geometrically, if v is a vertex of a rational billiard corresponding to a nonremovable singularity, when we sweep out an angle of 2π by continually reflecting the rational billiard in a side, the orientation is not preserved. In Fig. 6, we see that a vertex with an associated conic angle of 4π must sweep out an angle of 4π before returning to a copy with the same orientation. We calculate the conic angle of a conic singularity as follows. If, for j = 1, 2, ..., r, π(mj /nj ) is an interior angle of the rational billiard P , as in Proposition 2.3, then the corresponding conic angle is 2mj π. Consequently, the geodesic flow on SP does not see removable singularities and continues on unimpeded, but is immediately deterred from progressing across the surface when the geodesic intersects a nonremovable singularity. The geodesic flow on SP cannot be determined at such a singularity, implying that the billiard flow on P at the associated vertex cannot be determined either.10 Example 2.5. Consider again the triangle P (3π/8, π/8, π/2), as was done in Example 2.4. From the above formula, the conic angle of the conic singularity corresponding to the vertex with angle 3π/8 is 6π = 2 · 3π. Consequently, this conic singularity is of the nonremovable type. This means that the flow on the 10This implication follows from the equivalence of the billiard flow on P and the geodesic flow on SP .


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2 2

2 2 2

1

3

1

1

3

3

1

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3 3

2

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Figure 6. The vertex with angle measuring 2π/3 in the triangle above has a conic angle of 4π. Consequently, the only way to recover a correctly oriented triangle by way of reflecting through sides adjacent to this vertex is to sweep out an angle of 4π. surface SP cannot be determined through the corresponding copy of the vertex on SP . Moreover, we cannot logically define reflection at this vertex. The vertex at the center of our surface, however, is of the removable type; the corresponding conic angle is 2 · 1π = 2π; see Fig. 7. 2.4. KSn as a rational billiard. It was important that we developed an understanding of conic singularities, because, in some sense, the Koch snowflake is entirely comprised of singularities. Whether or not ‘corners’ of the Koch snowflake11 billiard can be shown to correspond to removable or nonremovable singularities of the flow remains to be determined. The prefractal billiard KSn is a rational billiard. Indeed, an interior angle of KSn is either equal to π/3 (acute) or 4π/3 (obtuse).12 Consequently, N = lcm{3, 3} = 3 and so the corresponding surface contains 2 · N = 6 copies of KSn . Furthermore, removable singularities of the surface correspond to acute angles of KSn and nonremovable singularities correspond to obtuse angles of KSn . This means that when α is a singular orbit of KSn , this orbit can be naturally extended to a periodic orbit if the singularities encountered in KSn were of the removable type; see Fig. 8. Likewise, if α is a singular orbit and the billiard ball encounters an obtuse angle of KSn , the flow on the billiard P cannot be naturally extended. In Fig. 9, we have represented the flat surface SKS1 , and in Fig. 11, we have depicted the generalized polygons corresponding to SKS1 , SKS2 and SKS3 . With sides properly identified, the latter generalized polygons become flat surfaces and can be viewed as the first three prefractal approximations of the ‘fractal flat surface’ (of infinite genus) SKS , which we will conjecture (in Section 4) to exist as a proper mathematical object; see part (i) of Conjecture 4.7. 11That is, vertices of the prefractal polygonal approximations KS , for any n ≥ 0. It is n worth noting that the union of these vertices for all n ≥ 0 forms a countable dense subset of ∂(KS). 12Furthermore, note that except for n = 0 (when KS = Δ, the equilateral triangle), we 0 always have both removable and nonremovable singularities (and in equal numbers).



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Figure 7. The surface SP corresponding to the triangle P (3π/8, π/8, π/2); the corresponding generalized polygon is an octagon, shown on the right. Again, we commit the sin of suggesting that the surface can be embedded in the plane, this time to illustrate the ambiguity experienced by the billiard ball at nonremovable singularities. Once the opposite sides of the octagon have been pairwise identified, as shown in the figure on the right, one obtains the surface SP . Note that periodic billiard orbits in the triangle (left) correspond to straight-line paths on the octagon (right), and hence to closed (i.e., periodic) geodesics on SP . This figure demonstrates that the two orbits emanating from points near a vertex corresponding to a nonremovable singularity behave in a radically different way; the two orbits are the two dotted lines, with the solid orbit representing what one would like to believe is the orbit emanating from the nonremovable singularity.

=⇒

Figure 8. Acute angles have associated removable conic singularities. Reflection at acute vertices can then be defined. We illustrate here a singular orbit of KS1 .


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Figure 9. The flat surface SKS1 corresponding to KS1 ; sides identified. This surface is independent of the flow and is the true surface, with sides properly identified. We cannot naively reflect in the adjacent sides of a fixed vertex and subsequently identify sides to produce the surface, because SKSn has genus g > 1 for all n > 0. There are six copies of the table KS1 in the associated flat surface, because 2 · lcm{3, 3} = 6. Moreover, these six copies are generated by letting the dihedral group D3 act on KS1 with an orientation designated by a vector that is not parallel to any of the sides; having such an orientation is a necessary condition for producing the correct number of copies of KS1 . In Fig. 10 below, we illustrate the associated straight-line flow on SKS1 for a particular periodic orbit of KS1 induced by the Fagnano orbit F0 of KS0 = Δ.

2.5. Equilateral triangle billiards and implications for KSn . The equilateral triangle billiard Δ is a very nice rational billiard. The billiard dynamics on Δ are well understood. Two orbits of an equilateral triangle billiard are said to be equivalent if each orbit has an unfolding that is a translate of the other and each have the same length. Denote the equivalence class of an orbit α by [α]. In [BaUm], Baxter and Umble give a classification of the periodic orbits of Δ, up to this notion of equivalence. Roughly speaking, they partition periodic orbits into two collections, those periodic orbits with odd period and periodic orbits with even



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Figure 10. The unfolding of the periodic orbit of KS1 induced by the Fagnano orbit F0 of KS0 = Δ.

Figure 11. The generalized polygons corresponding to KS1 , KS2 and KS3 , respectively; sides not identified. These should be viewed as the first, second and third level approximations of the limiting ‘generalized fractal polygon’ presumably associated with the billiard KS. When the proper identifications are made, such surfaces SKS1 , SKS2 and SKS3 can be viewed as the first, second and third level approximations of the limiting ‘fractal flat surface’ (as hypothesized in part (i) of Conjecture 4.7).


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Figure 12. The Fagnano orbit F0 of the equilateral triangle billiard Δ = KS0 . It is the unique shortest closed billiard trajectory of Δ and is given by the equilateral triangle inscribed in Δ. The associated initial conditions are (xmid , π/3), where xmid is the midpoint of one of the sides of Δ and π/3 is the corresponding initial angle of the path. period.13 Among orbits with even period, they then partition the orbits based on special criteria. If α is an even periodic orbit, then its equivalence class [α] has the cardinality of the continuum. The collection of odd period orbits is a collection of odd iterates of the Fagnano orbit F0 , {F02k+1 }∞ k=0 . Recall that the Fagnano orbit F0 is defined to be the shortest periodic orbit of the equilateral triangle billiard Δ;14 it is the unique such closed path and is represented in Fig. 12. Specifically, for every k ≥ 0, there is only one element of the equivalence class [F02k+1 ], that element being F02k+1 , the (2k + 1)th repetition of F0 . Roughly speaking, the billiard KSn is comprised of many copies of Δn , an equilateral triangle billiard with side lengths measuring 1/3n . More precisely, if TΔn is a tiling of the plane by equilateral triangles with side lengths measuring 1/3n , then KSn can be embedded in TΔn ; see Fig. 13. Because of this observation and the fact that all orbits of Δ can be unfolded in TΔn (see Fig. 14), it is reasonable to expect that periodic and singular orbits of Δ extend naturally to KSn . In fact, Fig. 14 illustrates that an unfolded periodic orbit of KS1 is collinear with the unfolding of the Fagnano orbit F0 of Δ = KS0 . 3. Experimental Results We want to describe the ‘anatomy’ of the prefractal billiard KSn . For fixed n ≥ 0, KSn is just a rational polygon. However, as n changes, the boundary of our prefractal billiard KSn changes. We want to be able to talk more concretely about the parts of KSn that ‘sprout up’ and those that ‘die off’ as n increases. Parts of KSn that ‘sprout up’ are what we will call cells of KSn . The collection 13The period of a billiard orbit is the number of times which the pointmass collides with the boundary. Consequently, an even period orbit is one for which the number of collisions with the boundary is even, and similarly for an odd period orbit. 14The name “Fagnano” for the shortest orbit comes from the name of the Italian mathematician Giovanni Fagnano whose work dates back to 1775. Indeed, Fagnano researched the existence of the shortest inscribed polygons of a fixed polygon P .



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Figure 13. Consider a tiling of the plane by the equilateral triangle Δn with side lengths 1/3n . Denote this tiling by TΔn . Then we can see that KSn ⊆ TΔn . This observation is illustrated here for the case when n = 1.

Figure 14. As noted in the caption of Fig. 13, KSn ⊆ TΔn . If we consider a periodic orbit γn,i in γn as shown, then we see that γn,i unfolds to a straight line. This unfolding is collinear with an unfolding of the Fagnano orbit F0 of Δn . (See §3.1 for the definition of γn .) As in Fig. 13, the situation is illustrated here for the case when n = 1.


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of segments of KSn−1 that are removed to construct the cells of KSn will then be called the ghost of KSn−1 . The ghost of KSn−1 will not be present in KSn , but such language will help us discuss the behavior of a certain type of orbits. We want to stress that we will not be discussing orbits of KSn . Rather, we will be discussing what we call induced orbits of KSn . What we mean by ‘induced’ is that if (x0 , θ0 ) is an initial condition of an orbit of Δ, an induced initial condition will be (x0 , θ0 ), where x0 and x0 are collinear in the direction of θ0 and x0 is a point on KSn . The new initial condition (x0 , θ0 ) is what we call an induced orbit of KSn . The reason for such language is that an orbit of KSn eventually induces an orbit of KSn+k for some k ≥ 1. Such an orbit is induced when the billiard ball passes through a ‘ghost’ of KSn+k−1 into a cell of KSn+k . We will discuss this in detail in §3.1, but first we formally define what we have discussed thus far and give an illustration of these concepts in Fig. 15. Definition 3.1 (The anatomy of KSn ). n

(i) (Ghosts of KSn ). Let n ≥ 0 and {sn,i }3·4 i=1 be the collection of segments comprising the boundary ∂(KSn ) of the billiard KSn . Then, for 1 ≤ i ≤ 3 · 4n , the open middle third of the segment sn,i is denoted by gn,i and is called the ghost of the segment sn,i . Moreover, the collection Gn = n {gn,i }3·4 i=1 is called the ghost set of KSn . The segments gn,i are removed in order to generate KSn+1 ; see Figs. 15(a)–(c). (ii) (Cell of KSn ). Let n ≥ 1, 1 ≤ i ≤ 3 · 4n−1 and Gn−1 be the ghost set of KSn−1 . Consider the set Gn−1 ∪ KSn and the bounded region in Gn−1 ∪ KSn given by an equilateral triangle with side lengths measuring 1/3n . Then this bounded region is what we call a cell of KSn . We denote a cell of KSn by Cn,i ; see Fig. 15(d) (by definition, KS0 = Δ has no cell). (iii) (Ghost of a cell). Let n ≥ 1, 1 ≤ i ≤ 3 · 4n−1 and Cn,i be a cell of KSn . Then the ghost gn−1,i of the segment sn−1,i is called the ghost of the cell Cn,i . In other words, the ghost of the cell Cn,i is the middle-third segment gn−1,i of the segment sn−1,i of ∂(KSn−1 ) that is removed from sn−1,i in order to generate that portion of ∂(KSn ); see the caption of Fig. 15. Example 3.2. If we consider the base of the equilateral triangle with sides having unit length, then the ghost of the base s0,1 of Δ = KS0 is given by g0,1 = (1/3, 2/3); see Fig. 15(a). Definition 3.3 (Compatible sequence of orbits). For each n ≥ 0, let αn be a periodic orbit of KSn . Then, the sequence {αn }∞ n=0 is said to be compatible if each αn is induced by α0 , the initial orbit of Δ. We performed computer simulations of induced orbits of the billiard KSn for n = 1, 2, 3. As a result, we now proceed to pose, discuss and sometimes answer various research questions regarding the induced orbits of KSn , for n ≥ 0. 3.1. γn Orbits of KSn . We define γn to be the collection of periodic orbits of KSn with an initial condition of the form (xmid , π/3), where xmid is a midpoint of an arbitrary side of KSn . When π/3 is not a feasible direction, let π/3 be an angle relative to the initial side. Every orbit in γn can then be demonstrated to be an orbit induced by some periodic orbit of the equilateral triangle Δ; see Fig. 14 for the case when n = 1, and for an illustration of the orbits in γ1 , see Fig. 16.



(a) The ghost set of KS0 = Δ, denoted by G0 .

(b) The elements of the ghost set G0 are removed.

(c) Out of every side there ‘sprouts’ two segments, giving rise to KS1 .

(d) G0 ∪ KS1 . The arrows indicate the cells C1,i , 1 ≤ i ≤ 3 · 40 = 3, of KS1 .

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Figure 15. An illustration of Definitions 3.1(i)–(iii) in terms of KS0 = Δ and KS1 . The ghost of the segment s0,i , denoted by g0,i , is a middle-third segment of s0,i and is removed from s0,i so that we may construct the cell C1,i of KS1 . Then g0,i is referred to as the ghost of the cell C1,i of KS1 .

Figure 16. There are two orbits in γ1 .


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Figure 17. We see that as we construct KS2 from KS1 , the orbit of KS1 induces an orbit of KS2 . Then, constructing KS3 from KS2 , the induced orbit induces an orbit of KS3 . However, this need not always be the case. Certain orbits would remain unchanged as we construct KSn+1 from KSn for some n. Moreover, we can see that the trajectory in the first image passes through the midpoint of the ghost gk,i of a cell Ck+1,i of KSk+1 for k ≤ 1. This fact will be significant in §4. We want to know what constitutes a shortest orbit of KSn . Is the shortest orbit an element of γn ? If not, are all the orbits in γn of the same length? Can we justify calling an orbit “Fagnano” in some sense that keeps with the ‘shortest length’ meaning of the name? We now proceed to answer these questions by examining our experimental results. Out of this will come a clear definition of Fagnano orbit of KSn and a better understanding of the behavior of the orbits in γn . Let γ1,i ∈ γ1 . Consider the unfolding of γ1,i induced by the classic Fagnano orbit F0 of KS0 = Δ, and consider the orbit α induced by the initial condition (xmid , π/6). Then, the orbit α is clearly shorter than the orbit γ1,i . Now, let us also consider the orbit γ2,j ∈ γ2 induced by F0 . Examine the unfolding of γ1,i and the unfolding of an equivalent orbit that does not reside on the ghost of the initial side, but on some other one-third-segment of the initial side. Since equivalent orbits have the same lengths, we can see that this is also a periodic orbit of Δ that is shorter than γ2,j but still longer than α. From this, we deduce that no orbit in γ2 will qualify as a shortest orbit of KS2 . Consequently, for every n > 0, no element of γn is a shortest orbit of KSn . However, this does not preclude us from identifying the shortest orbits in γn ; see Fig. 17. Consider the subcollection of γn comprised of orbits with initial point xmid ∈ Δ ∩ KSn ; see Fig. 18. In general, our results indicate that this subcollection is the collection of orbits with shortest length among all orbits in γn . We denote this subcollection by Fn and call it the collection of Fagnano orbits of KSn . Recall from our earlier discussion in §2.5 that F0 consists of a single element, namely, the Fagnano orbit (also denoted F0 ); see Fig. 12. We give a special name to the periodic orbit induced by F0 in the prefractal billiard KSn (for each fixed n ≥ 0). We call this particular orbit the primary piecewise Fagnano orbit of KSn (denoted by ppFn ) because, for every k ≤ n, the initial segment enters a cell Ck,i of KSk (by passing through the ghost gk−1,i of the



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Figure 18. From left to right: Δ ∩ KS0 , Δ ∩ KS1 , Δ ∩ KS2 , Δ∩KS3 . For each approximation KSk with k ≤ 3, we are removing a subset of the ghost set Gk of KSk . The set Δ ∩ KS is comprised of three ternary Cantor sets. cell Ck,i , as defined in Definition 3.1(iii)) and, upon reflecting, subsequently forms a true Fagnano orbit in a cell Ck . In general, based on this terminology, we refer to an orbit in γn as a piecewise Fagnano orbit. In Section 4, we will often denote the collection of piecewise Fagnano orbits of KSn by pFn (instead of γn ). 3.2. Nonsingular non-γn periodic orbits of KSn . Our characterization of periodic orbits of KSn in terms of what we have referred to as “γn ” and “non-γn ” orbits is by no means a rigorous classification of the induced orbits of KSn . Our results indicate that elements in γn are not always induced by the Fagnano orbit of Δ and non-γ0 orbits of Δ do not always induce elements in γn ; see Fig. 19. Since non-γ0 orbits of Δ can induce γn orbits of KSn , is it possible that non-γ0 orbits of Δ also induce singular orbits of KSn ? Can we demonstrate that there is some nonsingular non-γn orbit that induces a nonsingular periodic orbit of KSn+k , for all k ≥ 1? If n = 0 and x0 = xmid , then α(x0 ,π/3) , the periodic orbit of KSn with initial (or ‘inducing’) condition (x0 , π/3), is certainly a permissible element of the collection of non-γn orbits of KSn . If, for some integers j, ν ≥ 1 such that 0 < ν < 3j , we let x0 = (ν/3j , 0) on the base of Δ, then α(x0 ,π/3) induces a singular orbit of KSn for all n ≥ j; see Fig. 20. Moreover, this singular orbit collides with an obtuse angle of KSn , meaning that we cannot make sense of the billiard flow with this particular initial condition. However, experimental results are indicating that for an initial condition (x0 , θ0 ) = (xmid , π/6), the resulting orbit (i) will be periodic and (ii) may be nonsingular, for all n ≥ 0; see Fig. 21. With regards to the billiard Δ, changing the basepoint of the initial condition (x0 , θ0 ) will not affect the periodicity of the orbit, but may affect the length of the orbit. In our experiments regarding orbits in γn , we observed that changing the basepoint did not affect the periodicity of the orbit or the length of the orbit. This can be seen by unfolding the orbits. However, our experiments on non-γn orbits indicated otherwise. One would like to see that orbits emanating from a side sn,i with the same direction are equivalent, but this did not turn out to be the case. In particular, an example of a nonsingular non-γn periodic orbit that increased in length after a translation of the basepoint is one given by the inducing condition (xmid , π/6); see Fig. 21. Unlike the case of γn -type orbits, if we shift the basepoint of the inducing condition by some small perturbation δx, then the resulting orbit is longer in length; see Fig. 22.


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Figure 19. In this figure, we see an example of a non-γn periodic orbit of KS0 , KS1 , KS2 and KS3 , where we consider the same inducing condition (xmid , θ0 ) in each iteration of the prefractal billiards. For a fixed inducing condition (x0 , θ0 ), other simulations (not shown) are suggesting that induced orbits of KSn are (qualitatively) quite convoluted.

3.3. Singular orbits of KSn . Recall that a singular orbit of Δ is an orbit which collides with a vertex of the boundary ∂Δ. As we discussed in §2.3, vertices of a billiard with associated conic angles measuring 2π radians are considered removable singularities of the geodesic flow on SP . Since every vertex in Δ has an associated removable conic singularity, the billiard flow may be naturally defined at corners of Δ. Let α be an induced singular orbit of KSn . Since α is a singular orbit of KSn , α collides with either an obtuse angle or an acute angle of KSn . Recall that acute angles of KSn constitute removable singularities of the billiard flow whereas obtuse angles of KSn constitute nonremovable singularities of that flow. Consequently, we may not be able to always determine how α behaves at vertices of KSn . As n increases, KSn grows. Therefore, we may ask whether or not an induced singular orbit α remains singular as n increases. Because α can only be naturally extended at an acute angle vertex of KSn , is it possible to prove that induced singular orbits always collide with acute angles? Moreover, do singular orbits which only collide with acute angles at some stage n do so at every later stage n + k, k ≥ 1? Even



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Figure 20. The billiard flow cannot be determined at obtuse angles. However, examination of the corresponding geodesic flow on SKSn for n = 1, 2 allows us to draw ‘possible’ paths; note the ambiguity at obtuse angles. It is intriguing that the ‘possible’ choices that the billiard ball ‘makes’ at the triadic points {1/3, 2/3} and {1/9, 2/9} are exactly the same. This observation may allow us to force a logically consistent definition of reflection at nonremovable singularities of the billiard map on KSn . In other words, this figure suggests that there may be a rigorous way of reducing the ambiguity the billiard ball experiences at obtuse angles of KSn . Because of the unique ‘infinite’ symmetry enjoyed by the snowflake boundary ∂(KS), we may be able to express the billiard flow on KS at nonremovable singularities in terms of a (countably) infinite number of ‘possible’ paths the point mass may take. Again, we may be aided in this investigation by considering the (yet to be determined) geodesic flow on the associated ‘fractal flat surface’ SKS ; see Conjecture 4.7.

though our attention is focused on induced orbits, is it possible that there exist singular orbits of KSn that are not induced by any orbit of Δ? If α is an induced singular orbit of KSn , then experimental results indicate that α will always have to collide with a vertex of KSn+k , for all k ≥ 0. An explanation for why this may be so is that KSn+k can be embedded in TΔn+k and the inducing initial condition gave rise to a singular orbit of Δ. However, our experimental results show that there is no consistency in which types of vertices are encountered by the singular orbit. In particular, it can happen that a singular orbit of KSn may only collide with acute angles, but the induced orbit of KSn+k will collide with an obtuse angle for some k ≥ 1 and thus end there; see Fig. 23 for a concrete example.


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Figure 21. In this figure, we see an example of a non-γn periodic orbit of KS0 , KS1 , KS2 and KS3 . We consider the same inducing condition (xmid , π/6) in each iteration of the prefractal billiards. We note that because KSn can be embedded in TΔn (see Fig. 13), it is reasonable to expect that every singular orbit of KSn is induced. 3.4. Quasiperiodic orbits of KSn . When one considers a rational billiard P , the natural question to ask is “What are the periodic orbits of P ?” One usually does not consider initial conditions that do not give rise to periodic orbits.15 Moreover, if one has identified periodic orbits of P , aperiodic “approximations” to these orbits may not be worth analyzing. Nevertheless, when the boundary is changing— as is the case in the recursive construction of the Koch snowflake billiard table KS via the rational polygonal billiard tables KSn —such approximations, however such a notion is defined, may be interesting. We call these approximate orbits quasiperiodic orbits. Recall that every orbit of a billiard is assumed to have a unit-speed parameterization. Formally, we define what we mean by a quasiperiodic orbit of KSn as follows. Definition 3.4 (Quasiperiodic orbits of KSn ). Fix T > 0, n ∈ N, and let Pn denote the collection of all periodic orbits of KSn . An orbit q of the billiard KSn is called quasiperiodic if given any > 0, there exists a finite partition of [0, T ], 15See, however, the well-known Veech dichotomy [Ve1–3] (as described, e.g., in [HuSc,MaTa,Vo]), according to which a direction is either periodic or uniquely ergodic in the rational billiard table.



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Figure 22. In this figure, we see an example of what happens when we shift the initial point from xmid to xmid + δx, for some small perturbation δx. We consider the same inducing condition (xmid +δx, π/6) in each iteration of the prefractal billiards. What is interesting in this particular example is that the billiard trajectory seems to want to enter into a new cell with each iteration of the prefractal billiard. This indicates that there may be a limiting object to consider, presumably, a billiard trajectory of KS. t0 = 0 < t1 < t2 < ... < tl = T , and periodic orbits pj ∈ Pn such that |q(t) − pj (t)| < , for all t ∈ [tj−1 , tj ] and j ∈ {1, 2, ..., l}.


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Figure 23. The induced singular orbit only collides with acute angles of KS1 . The same initial condition induces a singular orbit of KS2 , whereas this time the orbit collides with obtuse angles of KS2 . Insufficient numerical accuracy prevents us from investigating (via computer simulations) the induced singular orbit of KS3 . As the definition would indicate, quasiperiodic orbits exist in any rational polygonal billiard table, but may have special significance in the Koch snowflake billiard. In Fig. 24, we see an example of a quasiperiodic orbit of KS1 that is ‘shadowing’ the γ1 orbits of KS1 . In general, if we consider a rational approximation to the initial angle θ0 , we obtain a quasiperiodic orbit of KSn . If n is changing and the billiard ball is passing through the ghost of a cell Cn+k,i , then we want to have a way of discussing the ‘stability’ of the resulting induced quasiperiodic orbits of KSn+k , for all k ≥ 1. Definition 3.5 (Stability of a sequence of quasiperiodic orbits). Let q0 be a quasiperiodic orbit of Δ and p0 a periodic orbit shadowed by q0 over some subinterval [tj−1 , tj ] ⊆ [0, T ] (as in Definition 3.4). Then we say that the sequence {qn }∞ n=0 of quasiperiodic orbits of KSn induced by q0 is stable if the sequence {pn }∞ n=0 of periodic orbits induced by p0 has the property that qn shadows pn for every n ≥ 0.16 With regards to quasiperiodic orbits of KSn , there are two types of potential behaviors that we would like to investigate. Specifically, we are naturally led to ask the following two questions. Firstly, if we fix n ≥ 0 as well as a given periodic orbit p of KSn , is there a sequence of quasiperiodic orbits {qi }∞ i=1 of KSn that converges to p, and if so, in what sense? Secondly, for a fixed quasiperiodic orbit q of KSn , in what ways is the induced quasiperiodic orbit of KSn+1 qualitatively and quantitatively different from q? More precisely, if qn is a quasiperiodic orbit of KSn , n ≥ 0, with a fixed rational angle a/b independent of n, and {qn+k }∞ k=0 is a sequence of induced quasiperiodic orbits, is the latter sequence stable (in the sense of Definition 3.5)? Our experimental results thus far are indicating that we can answer all of these questions in the affirmative. If {ai /bi }∞ i=1 is a sequence of rational approximations of π/3, then Fig. 25 describes what happens as ai /bi → π/3. This suggests that 16Hence, in the sense of Definition 3.3 above, {q }∞ n n=0 is assumed to be a ‘compatible’ sequence of quasiperiodic orbits.



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Figure 24. The left image depicts the quasiperiodic orbit q. The next two images depict the behavior of q over two subintervals of time (t0 , t1 ) and (t1 , t2 ). In the middle image, the corresponding periodic orbit p1 would be the one induced by the initial condition (xmid , π/3) and the corresponding periodic orbit p2 would be induced by ((1/4, 0), π/3), with the base of the generating equilateral triangle Δ lying on the x-axis and the lower left vertex being located at the origin. for a fixed n, there is some notion of convergence of quasiperiodic orbits to a corresponding induced periodic orbit α of KSn . For example, in the present case of Fig. 25 (where n = 1), the corresponding sequence {qi }∞ i=1 of quasiperiodic orbits KS1 seems to be converging (in some suitable sense) to the primary piecewise Fagnano orbit ppF1 of KS1 . As n increases, Fig. 26 illustrates that the quasiperiodic orbits break up over the boundary, but that the induced sequence of quasiperiodic orbits appears to remain stable, in the sense of Definition 3.5. Furthermore, if we increase the time T , as given in Definition 3.4, then Fig. 27 shows that the same behavior occurs.

4. Conjectures and Open Problems We propose here several conjectures and open problems regarding the billiard KS. As we have done in the previous sections, we begin by first defining a few necessary terms. Our ability to investigate the proposed billiard KS has been limited by the finiteness of the computer. Hence, our conjectures deal mostly with “what happens in the limit”. However, it is not exactly clear what we mean by limit. Hence, when we say ‘limit’, we are assuming a suitable notion of limit, most likely (depending on the context) an inverse limit of some inverse system or the Gromov–Hausdorff limit of a sequence of complete metric spaces; see, respectively, [HoYo] or [Gr] for an introduction to these notions. Once a suitable notion of limit has been defined, we want to consider the (admittedly, possibly empty) collection of periodic orbits of the billiard KS, which we denote by P. However, our focus thus far has been on induced orbits of KSn . We define I to be the collection of induced periodic orbits of KS, and phrase


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Figure 25. This figure depicts what happens as the approximate angle θ0 becomes more and more accurate. The quasiperiodic orbit seems to dry up as the initial angle θ0 → π/3. Here, like elsewhere in the experimental results reported in §3.4, the initial angle θ0 is a rational approximation of the initial angle θ of a given periodic orbit p, obtained via the continued fraction expansion of θ.

Figure 26. As the snowflake “grows”, we see that the quasiperiodic orbit breaks up over the boundary, but does not devolve into an incoherent collection of segments. Rather, there is some sense of stability. That qualitative sense of stability is captured quantitatively in Definition 3.5.



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Figure 27. Comparing this to Fig. 26, we see that for fixed n ≥ 0, a quasiperiodic orbit of KSn fills more of KSn as T increases, where T is as stated in Definition 3.4. If we consider a sequence of induced quasiperiodic orbits {qn }∞ n=0 , with qn a quasiperiodic of KSn , then this figure, along with Fig. 26, illustrates that the stability of {qn }∞ n=0 (in the sense of Definition 3.5) is (qualitatively) independent of the time T . Note, however, that both the partitioning of [0, T ] into subintervals, and the time intervals over which the shadowing of pn by qn occurs, are in general dependent on n. our conjectures in terms of elements in I . As in Section 3, where we discussed the ‘anatomy’ of KSn , we now discuss the anatomy of the proposed billiard KS and the collection of induced orbits I . We define the footprint 17 of an orbit to be the points of the orbit that belong to the boundary of the billiard KS. We call ∞ the ghost set of KS the collection G = n=1 Gn , the union of all ghost sets Gn associated with KSn (see Definition 3.1(i)). Definition 4.1 (Footprint of an orbit). Let α be an orbit of a billiard P . Then, the footprint of α is the set of points of the orbit that belong to the boundary ∂P . Definition 4.2 (Ghost set of KS). If Gn is the ghost set of KSn , then the ghost set of KS is defined by G = ∞ n=0 Gn .

17This term is sometimes used in the literature on billiards; see, e.g., [BaUm].


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Definition 4.3 (Self-similar orbit). Let α be a periodic orbit of the proposed billiard KS. Then, α is said to be a self-similar orbit if its footprint is a self-similar subset of ∂(KS) ⊆ R2 .18 Note, in particular, that by construction, the sequence of primary piecewise Fagnano orbits {ppFn }∞ n=0 is compatible, in the above sense (see Definition 3.3). Our first conjecture asserts the existence of a very special periodic orbit of the proposed billiard KS. We denote the (presumed) ‘limit’ of the orbits ppFn by ppF , and call this the primary piecewise Fagnano orbit of KS. We can see from Fig. 28 that there is an iterated function system (IFS),19 denoted by F , which is producing n scaled copies of pieces of the orbit, and such that for any n ≥ 0, ppFn = i=0 F i (F0 ). Observe that for each finite n, the chronology20 of the orbit ppFn can be easily recovered. Furthermore, if we consider the fixed point attractor of the IFS,21 F

=

lim F n (F0 ),

n→∞

then this set F should correspond to the footprint of the proposed periodic orbit ppF of KS.22 Defining the orbit ppF to be F is not very satisfactory, however, because such a definition does not provide useful information on how to recapture any sense of chronological order. Accordingly, we make the following formal conjecture. Conjecture 4.4. Under a suitable notion of limit, we conjecture that the chronology of ppF can be naturally realized and hence, that ppF can be considered as a true periodic orbit of the Koch snowflake billiard KS. Significant analytical evidence in support of ppF being a well-defined orbit of the billiard KS is the fact that (i) ppF is a self-similar orbit (specifically, its footprint appears to be a self-similar Cantor set) and (ii) has finite length. Once we have established the existence of the periodic orbit ppF , we expect that demonstrating the existence of other periodic orbits given as ‘limits’ of compatible sequences of induced periodic orbits {αn }∞ n=0 may not be an unreasonable endeavor. Specifically, we conjecture that orbits emanating from the boundary of the proposed billiard KS at an angle of π/3 can be realized by a suitable generalization of the procedure (yet to be precisely determined) that would substantiate our claim that ppF forms a periodic orbit of the Koch snowflake billiard. Our next conjecture makes specific Conjecture 4.4 and the comments surrounding it. It also discusses some of the geometric properties of the presumed ‘piecewise Fagnano orbits’ of KS, including the ‘primary piecewise Fagnano orbit’ ppF , the existence of which was asserted in Conjecture 4.4. 18Here and thereafter, when talking about an IFS or a self-similar set, we are making the same slight abuse of language as in §1 when referring to the Koch snowflake curve ∂(KS) as being ‘self-similar’, whereas strictly speaking, ∂(KS) is obtained by fitting together three copies of a self-similar set, namely, the von Koch curve; see Figs. 2 and 3. 19More specifically, in the spirit of footnote 18, this ‘IFS’ is given by three different IFS’s, denoted by F1 , F2 , F3 and each comprised of two contractive similarity transformations of R2 ; see the caption of Fig. 28. The associated attractor is therefore a ‘self-similar set’ of R2 (also in the spirit of footnote 18). 20The chronology of an orbit is the order in which the billiard ball visits points of the boundary of the billiard table. 21See, e.g., [Fa] for a detailed discussion on iterated function systems. 22Note that by construction, F is a self-similar subset of ∂(KS) ⊆ R2 .



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Figure 28. Motivation for building the primary Fagnano orbit of KSn by means of an iterated function system (IFS): the IFS produces the orbit ppFn of KSn by first contracting, rotating and translating the Fagnano orbit of Δ, so that what results are three copies of F0 appropriately appended to F0 . In actuality, there are three IFS’s (F1 , F2 , F3 ) acting in unison to produce this picture. Specifically, F1 acts on the upper left scaled copy of F0 to produce two scaled, rotated and translated copies of that piece of ppF1 . Likewise, F2 acts on the upper right copy of F0 to produce two scaled, rotated and translated copies of that piece of the orbit ppF1 . Finally, F3 behaves similarly on the bottom scaled copy of F0 . The union of all three images unioned with ppF1 is then the orbit ppF2 . Continuing in this manner, we can produce ppFn , for every n ≥ 0. Conjecture 4.5 (Existence of self-similar periodic orbits of KS). (i) (The primary piecewise Fagnano orbit of KS). The periodic orbit ppF of KS can be realized (and its chronology restored ) as a suitable limit (possibly, an inverse limit) of the compatible sequence of primary piecewise Fagnano orbits ppFn of KSn . Furthermore, the periodic orbit ppF is a self-similar orbit (in the sense of Definition 4.3); more specifically, its footprint is the natural self-similar middle-third Cantor set, strung around the boundary ∂(KS). (ii) (Piecewise Fagnano orbits of KS). More generally, an arbitrary ‘piecewise Fagnano periodic orbit’ α of KS can be defined and realized as follows.


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There exists a compatible sequence {αn }∞ n=0 of periodic orbits such that for each n ≥ 0, αn belongs to pFn , and {αn }∞ n=0 converges (in a suitable sense) to α.23 Furthermore, α is a self-similar orbit of KS. The following open problem is motivated by our earlier discussion in §3.2 and §3.3.24 Open Problem 4.6. On the base of Δ = KS0 , does there exist x0 (other than the midpoint xmid ) such that the sequence {αn }∞ n=0 of compatible non-γn periodic orbits induced by the initial condition (x0 , π/6) will (i ) avoid all obtuse angle vertices of KSn , for every n ≥ 0 and (ii ) converge to a periodic orbit α of KS? Moreover, we conjecture that if the initial angle θ0 of the trajectory is not π/3, but such that (x0 , θ0 ) would have induced a nonsingular non-γn periodic orbit αn of KSn for all n ≥ 0, then the associated compatible sequence {αn }∞ n=0 ‘converges’ to a periodic orbit α of KS; and correspondingly, the associated sequences of footprints of {αn }∞ n=0 converges to the footprint of α. In addition, we conjecture that the footprint of α is a topological Cantor set.25 A primary candidate for illustrating this conjecture is the periodic orbit induced by the initial condition (xmid +δx, π/6), for some small perturbation δx; see Fig. 22. A key step towards placing the above conjectures and open problems in a broader context and on firmer mathematical grounds must involve the theory of flat surfaces associated with rational billiards, along with its eventual extension to surfaces of infinite genus (called here ‘fractal flat surfaces’) presumably associated with fractal billiards.26 Ultimately, we are conjecturing the existence of a suitably defined billiard table KS. This amounts to establishing that (i) there is a corresponding flat surface SKS and (ii) the billiard flow on KS is equivalent to the billiard flow on SKS . In particular, we conjecture that the ‘limit’ of the sequence of prefractal billiard tables {KSn }∞ n=0 would correspond to this billiard KS, and that similarly, the associated surface SKS would be the ‘limit’ of the sequence of prefractal flat surfaces {SKSn }∞ n=0 . If we can prove that there are removable singularities in the proposed surface SKS , then it is not unreasonable to expect the conjecture to be true. Moreover, if the wonderful symmetry of the Koch snowflake curve ∂(KS) can be exploited so as to provide us with a way of dealing with the ambiguity the billiard ball experiences at non-removable singularities (see, e.g., the caption of Fig. 20), then we would be further justified in believing the conjecture to be a plausible statement. We summarize the main aspects of this central conjecture—and clearly, longterm open problem—in the following more specific form. Naturally, part of the

23Recall from the end of §3.1 that pF stands for the collection of piecewise Fagnano periodic n orbits of KSn . 24Furthermore, it has an obvious counterpart in the case where x = x 0 mid is allowed to be on any of the sides of Δ. 25I.e., it is a perfect and totally disconnected subset of the boundary ∂(KS). 26The flat surface S of a rational polygonal billiard P always has a finite genus. Moreover, P a simple calculation based on the known formula for g(SP ) (see, e.g., Lemma 1.2, p. 1022 of [MaTa]) shows that gn = g(SKSn ) → ∞ as n → ∞. Hence, it is natural to expect that the presumed limiting surface SKS has infinite genus; see Conjecture 4.7.



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difficulty in dealing with this problem will involve finding the appropriate notions of limits involved in the formulation of the conjecture.27 Conjecture 4.7 (The fractal billiard KS and fractal surface SKS , along with the associated flows). (i) (The fractal flat surface SKS , along with the geodesic flow). The sequence {SKSn }∞ n=0 of prefractal flat surfaces associated with the rational billiard KSn converges (in the Gromov–Hausdorff sense, see [Gr]) to a surface of infinite genus SKS , called the ‘fractal flat surface’ associated with KS. Correspondingly, the ‘geodesic flow’ on SKS can be realized as a suitable limit (possibly, an inverse limit) of the geodesic flows on the surfaces SKSn . (ii) (The Koch snowflake fractal billiard KS, along with its billiard flow). The fractal billiard KS can be defined as a suitable limit (possibly, an inverse limit) of the prefractal polygonal rational billiards KSn ; essentially,28 this means that the billiard flow on KS can be realized as a suitable limit (also possibly, an inverse limit) of the corresponding billiard flows on the billiard tables KSn . (iii) (Geodesic vs. billiard flow). Finally, the geodesic flow on the fractal flat surface SKS is equivalent to (and, at first, may be used as a suitable substitute for ) the billiard flow on the Koch snowflake billiard KS. We close this paper by stating the following very long-term problem, which is directly motivated by the questions raised in [La2] concerning the relationship between ‘fractal drums’ and ‘fractal billiards’, including the Koch snowflake drum and billiard. See, in particular, [La2], Conjecture 6, p. 198, itself motivated by Conjectures 2 and 3, pp. 159 and 163–164, respectively; see also §12.5.3 of [La-vF]. For information regarding the Koch snowflake drum and other ‘fractal drums’ (viewed as ‘drums with fractal boundary’), see, e.g., [La1–2,LaNRG,LaPa], [La-vF,§12.3 & §12.5], and the relevant references therein. For information regarding trace formulas (including the Gutzwiller and Chazarain trace formulas) in various contexts, we refer, e.g., to [Gz1,2], [Ch], [Co] and [DuGn]. Open Problem 4.8 (Fractal Billiard vs. Fractal Drum.). Once the existence of the Koch snowflake billiard KS has been firmly established (as hypothesized in Conjecture 4.7), can one formulate, and eventually establish, a suitable ‘fractal trace formula’ in this context? Presumably, the latter would be a fractal counterpart of the Gutzwiller, Chazarain and the Selberg trace formulas in this context, connecting the length spectrum of the snowflake billiard29 and the eigenvalue (or frequency) spectrum of the corresponding snowflake fractal drum. Moreover, can one address 27For example, even if the notion of ‘inverse limit’ is suitable for formulating part (ii) or the

end of part (i) of the conjecture, one would still need to specify the maps involved in the definition of the underlying inverse systems; see, e.g., [HoYo]. 28It is clearly true (and well known) that viewed as a sequence of compact subsets of R2 , the billiard tables KSn converge in the sense of the Hausdorff metric (or, equivalently, in the Gromov–Hausdorff sense) to the Koch snowflake billiard table KS. Hence, the real issue concerns here the associated billiard flows. 29Or, more accurately, the collection of (suitable equivalence classes of) periodic orbits of the billiard flow of KS (i.e., essentially equivalently, of the geodesic flow of SKS ; see part (iii) of Conjecture 4.7).


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the same problem for other fractal billiards (once they have been properly defined ) and the associated fractal drums? We hope that the study of such open problems and conjectures, aided by a suitable combination of computer experiments and theoretical investigations, will enable us in the future to better understand the elusive nature of the Koch snowflake billiard and, eventually, of a variety of other fractal billiards. Added note. The present paper can be found on the Mathematics arXiv (arXiv:0912.3948) with color pictures. Acknowledgements. We wish to thank Pascal Hubert for his helpful comments on a preliminary version of this paper. References [BaUm] [Ch] [Co] [DuGn] [Fa] [GaStVo] [Gr] [Gu] [GuJu1] [GuJu2] [Gz1] [Gz2] [HoYo] [HuSc]

[KaHa] [KaZe] [La1]

[La2]

A. Baxter and R. Umble, Periodic orbits of billiards on an equilateral triangle, Amer. Math. Monthly No. 8, 115 (2008), 479–491. J. Chazarain, Formule de Poisson pour les variét´ es riemanniennes, Invent. Math. 24 (1974), 65–82. Y. Colin de Verdi` ere, Spectre du laplacien et longueur des g´ eod´ esiques périodiques, I et II, Compositio Math. 27 (1973), 83–106 and 159–184. J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39–79. K. J. Falconer, Fractal Geometry: Mathematical foundations and applications, John Wiley & Sons, Chichester, 1990. G. Galperin, Ya. B. Vorobets and A. M. Stepin, Periodic billiard trajectories in polygons, Russian Math. Surveys No. 3, 47 (1992), 5–80. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Modern Birkh¨ auser Classics, Birkh¨ auser, Basel and Boston, 2001. E. Gutkin, Billiards in polygons. Survey of recent results, J. Stat. Phys. 83 (1996), 7–26. E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett. 3 (1996), 391–403. E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J. 103 (2000), 191–213. M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys. 12 (1971), 343–358. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics, vol. 1, Springer-Verlag, New York, 1990. J. G. Hocking and G. S. Young, Topology, Dover Publ., Mineola, 1988. P. Hubert and T. Schmidt, An introduction to Veech surfaces, in: Handbook of Dynamical Systems, vol. 1B (A. Katok and B. Hasselblatt, eds.), Elsevier, Amsterdam, 2006, pp. 501–526. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge, 1995. A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes 18 (1975), 760–764. M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl–Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), 465–529. M. L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl–Berry conjecture, in: Ordinary and Partial Differential Equations (B. D. Sleeman and R. J. Jarvis, eds.), vol. IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Math. Series, vol. 289, Longman, Scientific and Technical, London, 1993, pp. 126–209.



[LaNRG]

[LaPa] [La-vF]

[Ma] [MaTa]

[Sm]

[Ta] [Ve1] [Ve2] [Ve3] [Vo] [Zo]

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M. L. Lapidus, J. W. Neuberger, R. J. Renka and C. A. Griffith, Snowflake harmonics and computer graphics: Numerical computation of spectra on fractal domains, Internat. J. Bifurcation & Chaos 6 (1996), 1185–1210. M. L. Lapidus and M. M. H. Pang, Eigenfunctions of the Koch snowflake drum, Commun. Math. Phys. 172 (1995), 359–376. M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and spectra of fractal strings, Springer Monographs in Mathematics, Springer-Verlag, New York, 2006. (Second revised and enlarged edition to appear in 2010.) H. Masur, Closed trajectories for quadratic differentials with an applications to billiards, Duke Math. J. 53 (1986), 307–314. H. Masur and S. Tabachnikov, Rational billiards and flat structures, in: Handbook of Dynamical Systems, vol. 1A (A. Katok and B. Hasselblatt, eds.), Elsevier, Amsterdam, 2002, pp. 1015–1090. J. Smillie, Dynamics of billiard flow in rational polygons, in: Dynamical Systems, Encyclopedia of Math. Sciences, vol. 100, Math. Physics 1 (Ya. G. Sinai, ed.), SpringerVerlag, New York, 2000, pp. 360–382. S. Tabachnikov, Billiards, Panoramas et Synth` eses, vol. 1, Soc. Math. France, Paris, 1995. W. Veech, Teichm¨ uller geodesic flow, Annals of Math. 124 (1986), 441–530. W. Veech, Teichm¨ uller curves in modular space, Eisenstein series, and an application to triangular billiards, Invent. Math. 97 (1989), 553–583. W. Veech, Flat surfaces, Amer. J. Math. 115 (1993), 589–689. Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative, Russian Math. Surveys 51 (1996), 779–817. A. Zorich, Flat surfaces, in: Frontiers in Number Theory, Physics and Geometry I (P. Cartier, et al., eds.), Springer-Verlag, Berlin, 2002, pp. 439–585.

Department of Mathematics, University of California, Riverside, CA 92521–0135, USA E-mail address: [email protected] Department of Mathematics, University of California, Riverside, CA 92521–0135, USA E-mail address: [email protected]
